Techniques for the Classroom
As mentioned early on in this unit, I want my students to see numbers conceptually and pictorially, and also to understand the algorithm, all of which are done in the Singaporean approach to mathematics. There are resources that I use in my classroom that help me to achieve this goal with my students. The order I recommend teaching them in the classroom, depends on whether you're working with a conceptual, pictorial, or abstract example. Thus, one technique below may be more illustrative than another.
Cuisenaire Rods
Cuisenaire rods were created by Georges Cuisenaire in 1952, and are a great way to represent the base-ten number system. I like to use this technique at the beginning stages of the base-ten number system. Cuisenaire rods come in varying colors and each color represents a certain number of units. The rods typically available are: white (1), red (2), light green (3), lavender (4), yellow (5), dark green (6), black (7), brown (8), blue (9), and orange (10). I spend time developing how much each rod is worth and do so by for example, taking a yellow rod and having students figure out how many white units would be needed to create a yellow. I chose white because they are worth one unit, so relevant to this unit's focus on place value and the decomposition of numbers into their base-ten components. It is important to emphasize this point to your students as well once students have an understanding of this, I would present computation problems to them and allow them to use Cuisenaire rods to complete the computation. So, for example a red rod (2) and a black rod (7) together and would add up to 9. Or, and orange rod (10) could be placed above a red and black and students could be asked how many more are needed to make ten. Students should place a white (1) with the red and black. This would help them to see that one more is needed to make ten.
Number Bonds
Numbers bonds were first used in the Singapore math curriculum in the late 1970s. Number bonds are a great activity that can be completed daily to help students build flexibility with adding single digit numbers. They allow students to think of all the numbers that bond together to make a specific number. What I do is present my students with various numbers. For instance I will say "What numbers bond with 8?" They should respond with:
Working with number bonds allows students to become flexible with numbers and realize that there is more than one way to represent a particular number.
Base-Ten Blocks
Base-ten blocks consist of four basic blocks: units (ones), long (tens), flats (hundreds), and blocks (thousands). Many people believe that base-ten blocks and Cuisenaire rods are the same however, they are not. They achieve the same purpose, but they are two different tools. I feel that Cuisenaire rods allow for students to see the decomposition of numbers a little bit better than base-ten blocks. However, base-ten rods allow students to represent the tens and ones in numbers better than Cuisenaire rods. Having said that, "Base-ten blocks should come later in the learning process, once a student has gained a firmer understanding of the base-ten system." This conclusion was gathered from conversations that I have with fellow teachers who have used the base-ten and Cuisenaire rods already. The long pieces allow them to grasp that a ten is composed of ten ones quickly because it is segmented into ten sections that can not be broken apart. I believe that base-ten blocks allow students to add their numbers with ease. Students can also add and subtract with Cuisenaire rods, but it may be more difficult because of all the different combinations. Once your students understand the decomposition of two digit numbers, I encourage you to introduce base-ten blocks.
Number Cards
Number Cards are a great way to illustrate the teen numbers. As mentioned previously the teen numbers are weird and this can cause problems with a student's understanding of exactly what each number illustrates. Number cards help to bring some meaning to these weird numbers. For example, in understanding the number 12 students need to understand that it is composed of one ten and two ones. To illustrate this with number cards, I would have a 10 written on one card and a 2 written on another card. I would place the 10 down first and place the 2 overlapping the 10 just a little.
Presenting the numbers in this way allows students to see that twelve is one ten and two ones. The visual that the cards present is phenomenal and for some learners really helps them to understand the teen numbers. It is also a great way to illustrate how numbers bond together. This same process can be completed with the "ty" numbers as well.
Ten Frames
The purpose of ten frames is to get students thinking about numbers in relationship to 10. This tool allows students to see place value easily. Ten frames are arrays made of two rows with five columns.
This illustration represents the number four on the ten frame. For example, I might say "If I have a ten frame with six dots, how many more do I need to make ten?" The response my students should supply is 4. I can also show them a ten frame with one dot and ask "How much less than 10 is my ten frame?" The response should be nine. The different concepts don't have to be taught in isolation, but can be made to play off of one another once students have a firm understanding of the concept that is being presented. Following the activities in this unit, you will find a sample worksheet containing 14 ten frames.
Number Line
The number line is a straight horizontal line with a prescribed origin (labeled "0") and a prescribed unit length (labeled "1"). On the line, the other integers are labeled according to this prescribed unit length, origin, and positive direction. That is, 2 would be marked to the right of 1, and should be the same distance away from 1 as 1 is from 0. This process continues with positive integers continuing to the right of 0, and negative integers to the left.
Students can use this line to help them add numbers. For example, if they have the problem 43 + 12 they would start at 43 and first move ten to the right and land on 53. Then they will move two more spaces to the right and land on 55. If students have a firm understanding of this line, they can also transfer the skills used for adding numbers on this line to measuring objects in the real world.
Introduction to Activities
I will be sharing three activities with you. Each of the activities gives examples of one of the three aspects that were mentioned earlier in my research paper: conceptual, pictorial, and abstract. I would like to reiterate the purpose of each of these steps again. During the conceptual stages, students are trying to understand the concept of tens and ones. After grasping the conceptual stage, the pictorial stage begins. The pictorial stage requires pictures to be drawn to illustrate the tens and ones. The final stage is the abstract, and during this stage, the algorithm is introduced. Each of the activities presented will focus on one of the three stages just mentioned. It may appear the activities are similar, but they are different because they are exposing the students to different stages of the process.
Activity One – Number of Days (Conceptual)
Objective: Students will understand decomposition of numbers.
Materials: Cuisenaire rods, Area to display work created for number of days in school
Procedure:
1. Establish a place to keep track of the number of days you have been in school.
2. Keep a running total of the number of days school has been in session. I like to keep mine on a place value chart similar to the model below:
3. Make sure you have Cuisenaire rods.
4. Represent the number of days you have been in school with Cuisenaire rods. Emphasize the different ways to create the number of days school has been in session with Cuisenaire rods.
If we have been in school 14 days, I should have a ten rod to represent 10 and then the students could have:
Ask students questions with regard to the number of day you have been in school.
Some examples are:
How many more days do we have to be in school to reach 20 days (Next landmark number)? The answer given our example would be 6 more days. Again have the students represent this with Cuisenaire rods:
Show me this number with Cuisenaire rods. (Make sure all possibilities are shown with the Cuisenaire rods.)
5. Allow students to return to their seats and supply five or more equations that
equal the number of days we have been in school. Some sample equations are:
10 + 4 = 14
6 + 4 +4 = 14
20 – 6 = 14
15 – 1 = 14
14 x 1 = 14
Activity Two – Ten Frames (Pictorial)
Objective: Students will use ten frames to help with adding and subtracting numbers.
Materials: Ten Frames (See Ten Frame Sheet) Make Multiple Copies
Procedure:
1. Make sure you make copies of the ten frames for your students.
2. Remind your students that the number system we use in the United States is the base-ten number system. The reason it is called base-ten is because it is composed of ten numbers.
3. Give them a problem. 46 +16. Allow your students to use the ten frames to solve the problem. First, students should decompose the numbers 46 and 16 into tens and ones: 46 = 40 + 6, and 10 + 6. Next, students should add the ones (6 + 6) together.
Here is 6 + 6 on the ten frames:
Your students should realize that 6 + 6 = 12 (one frame is filled completely and
the other has two that are filled.
4. Next students should use the ten frames to add the (40 + 10) together as follows:
Students should see from this illustration that 40 + 10 = 50 because you have
five ten frames that are full. Looking at the tens and ones yours students should
see six ten frames that are full another ten frame containing only two dots, so the
total that they are left with is 46 + 12 = 62.
Activity Three – Adding Tens and Ones (Abstract)
Objective: Students will add and subtract numbers with tens and ones.
Materials: Paper
Procedure
1. Make sure students have a firm understanding of decomposing numbers.
2. Present the problem to the students: 25 + 14 =
3. Decompose 25 = 20 + 5, and 14 = 10 + 4
4. Students should add ones 5 + 4 = 9
5. Students should add tens 20 + 10 = 30
6. Add the ones and tens together to get the answer 9 + 30 = 39.
7. If they have the subtraction problem: 36 – 23 =
8. Decompose 36 = 30 +6, and 23 = 20 + 3
9. Students will subtract the ones first 6 – 3 = 3
10. Students will subtract their tens 30 – 20 = 10
11. Students will add the tens and ones together 3 +10 = 13
12. Present the problem to the students 53 – 19 = (This problem requires borrowing.)
13. Decompose 53 = 40 + 13, and 19 = 10 + 9
*Notice how I moved a ten into my ones as 10 ones. This get me ready for the next
step.
14. Students will subtract the ones 13 – 9 = 4
15. Students will subtract the tens 40 – 10 = 30
16. Students will add the tens and ones together 4 + 30 = 34
Comments: